When you analyze rational expressions, you often have to factor out polynomials to find a solution. For example, the rational expression (x^2 + 10x + 25)/(x^2 - 25) becomes (x + 5)^2/(x + 5)(x - 5), or (x + 5)/(x - 5). Such processes as the solution of matrices in Algebra II rely on a knowledge of factoring to take place.
When dealing with rational expressions, students learn how to combine like terms after multiplication using the FOIL method or after factoring as in the prior section. Like terms have not only the same variable but also the same degree of each variable; for example, 3x^3 and 5x^2 are not like terms and are not able to be combined.
The ability to eliminate different items from expressions can make the difference between a simple problem and an incredibly complex one, and between the correct answer and one that is far off the mark. Hopefully, you wouldn't simply (3 + 6)/3 to be 6/1 by taking out the threes, because the operation is different in the denominator. However, many students often start out with rational expressions by simplifying (x + 5)/x to 5/1 because of a similar error.
Using reciprocals and adding fractions with different denominators is difficult enough for students when numbers are involved. With different variables and different exponential degrees, these manipulations can be even more difficult. Learning to follow such rules as P/Q + R/S = (PS + RQ) / QS and gaining familiarity with the concept that 1 / P / Q = Q / P will serve students well when even more complicated problems show up later in high school and in college.